Stability and shapes of cellular profiles in directional solidification: expansion and matching methods

Ideas based on constitutional supercooling suggest that the periodic steady state cellular patterns seen in the directional solidification of systems with small partition coefficient may be unstable if the impurity concentration in the melt just in front of the tips falls into the two phase (miscibility gap) region of the phase diagram. This gives a simple stability criterion relating the position of the tips of the cells to the pulling velocity that is in good qualitative agreement with the limited experimental data available in the cellular regime. Implications of this criterion for a particular class of steady state solutions derived using asymptotic matching methods are explored. These solutions arise from a generalization to finite Péclet number for systems with small partition coefficient of the ideas of Dombre and Hakim relating directional solidification patterns to viscous (Saffman—Taylor) fingers. Families of steady state solutions yielding both small amplitude interface patterns as well as fingerlike solutions with narrow deep grooves are accurately described by the methods discussed herein. A systematic expansion method provides corrections to the classical Scheil shapes for the grooves. However, the stability criterion, as well as other considerations, suggest that the entire class of narrow grooved solutions found by the matching methods may be unstable. Comparison with other numerical work suggests that other branches of narrow grooved solutions exist and are relevant to experiments. Several experimental and numerical tests of these ideas are proposed.